Subgroups — ΓL(1,𝔽₈)×C₇₅₁ #30628828

Order 15771.

8 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₇₅₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₇₅₁	—	7	1	no	maximal, Hall
3	C₇₅₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
4	ΓL(1,𝔽₈)	C₇₅₁	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₇	C₃×C₇₅₁	1	1	yes	characteristic, complemented, Sylow
6	C₃	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇₅₁	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃×C₇₅₁
Frattini	C₁	G
Fitting, socle	C₇×C₇₅₁	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₇₅₁

Splittings G = N ⋊ H with N normal; H ≅ G/N.

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.